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Difficulty: MediumAlgebraic Word Problems and Modeling

A logistics company operates two delivery vans, Van A and Van B. Van A consumes fuel at a rate of 11 gallon for every 1515 miles driven, and Van B consumes fuel at a rate of 11 gallon for every 2525 miles driven. On a specific trip, the two vans were driven a total combined distance of 450450 miles and together consumed exactly 2222 gallons of fuel. What was the total distance, in miles, driven by Van A?

  1. A
    100 miles
  2. 150 milesAnswer
  3. C
    225 miles
  4. D
    250 miles
  5. E
    300 miles

Answer

The total distance driven by Van A was 150 miles.
The correct answer is 150 miles. By setting up the total fuel consumption equation as dA15+450−dA25=22\frac{d_A}{15} + \frac{450 - d_A}{25} = 22, multiplying by the least common multiple 7575 gives 5dA+1350−3dA=16505d_A + 1350 - 3d_A = 1650, which simplifies to 2dA=3002d_A = 300 and dA=150d_A = 150 miles.

Step-by-Step Solution

1
Define variables for the distance traveled by each van.
Let dAd_A be the distance driven by Van A. Then the distance driven by Van B is 450−dA450 - d_A.
The total combined distance for both vans is 450450 miles.
2
Express the total fuel consumed in terms of dAd_A.
\frac{d_A}{15} + \frac{450 - d_A}{25} = 22
Fuel used equals distance divided by miles per gallon for each vehicle.
3
Clear denominators by multiplying the entire equation by the common denominator 75.
5d_A + 3(450 - d_A) = 1650
Simplifies fractions to solve the linear algebraic equation easily.
4
Solve for dAd_A.
5d_A + 1350 - 3d_A = 1650 \implies 2d_A = 300 \implies d_A = 150
Isolates the target variable dAd_A to find the distance driven by Van A.

Key Concept

Linear Equations and Rate-Distance Modeling
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Topics

Algebraic Word Problems and Modeling

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